Objectivism: fascinating but flawed.

I've always been interested in all kinds of philosophy, studying them on and off as I came into touch with new ideas. As of lately, Objectivism, Ayn Rand's philosophy has been becoming somewhat more popular. I also looked into it, and while it's a fascinating attempt to build an objective basis for philosophy, knowledge, ethics based on a few basic axioms, I also see that it is flawed in several ways.

Rand's philosophy begins with three axioms: existence, identity, and consciousness, and ends up with egoism and laissez-faire capitalism as ethical ideals. In many ways, she and her followers are applying logic in incorrect ways, and use terms in vague ways, which ends up producing conclusions that do not follow from the premises. Furthermore, they seem to be unaware of two traps in constructing knowledge of reality (which I will assume as "real" for the time being).

The first trap is the trap of the sensations. Basically, our awareness of reality is nothing but a whole of sensations. But we cannot assume that a sensation accurately represents reality. An photo of a pipe is not a pipe, even though they produce somewhat similar visual sensations.

The second trap is the trap of logic and math. Logic and math can be used to derive true knowledge from true premises, but only if the logic and math we use are applicable to the problem. You see, math is simply a rigorous game of thought, where we start with definitions, use symbolic notation, and start with a couple of axioms, that is statements assumed to be true, and then construct a system out of that. 

However, in the same subject matter, different axioms are possible. In geometry, we have Euclidian (planar) geometry, hyperbolic geometry and spherical geometry, each of which have ony one different axiom between themselves, but come to vastly different conclusions from them. When have need of geometry, we must then keep into consideration which geometry is applicable to our problem. This we can only do by validating our axioms in an external way. A pilot of an airplane uses spherical geometry as the earth has been proven in other ways to be spherical. But for an architect, Euclidian geometry is more convenient.

Logic is a form of math too, and suffers from the same problem. There is not a single system of logic, rather, there are many. The question then becomes, which logic to use? Of course, we want the logic that works correctly for understanding reality, but we can't just pick one and assume that it does so!

In there lies the core of the challenges to obtaining true knowledge of reality.